A338593 -id:A338593 - cp's OEIS frontend

A052448 Number of simple unlabeled n-node graphs of edge-connectivity 3.

0, 0, 0, 1, 2, 15, 121, 2159, 68715, 3952378, 389968005, 65161587084

Offset: 1

Eric W. Weisstein, May 08 2000

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Jens M. Schmidt, Combinatorial data.
Eric Weisstein's World of Mathematics, k-Edge Connected Graph.

Column k=3 of A263296.
Cf. other edge-connectivity unlabeled graph sequences A052446, A052447, A241703, A241704, A241705.
Cf. A338511, A338583, A338593, A338594, A338604, A339072.

a(8), a(9), a(10) from the Encyclopedia of Finite Graphs by Travis Hoppe and Anna Petrone, Apr 22 2014
a(11) by Jens M. Schmidt, Feb 18 2019
a(12) from Jens M. Schmidt's web page, Jan 10 2021

A338594 Number of unlabeled connected planar graphs with n edges with degree >= 3 at each node.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 17, 37, 98, 275, 797, 2414, 7613, 24510, 80721, 270018, 915034

Offset: 6

Hugo Pfoertner, Nov 21 2020

			a(6) = 1: the 3-connected edge graph of the tetrahedron;
a(7) = 0: no connected planar graph with degree >=3 at each node exists;
a(8) = 1: the 3-connected 5-wheel graph, edge graph of 4-sided pyramid;
a(9)-a(11): see linked illustrations.

Hugo Pfoertner, Illustrations of terms a(9) - a(11).

Cf. A007112, A002840, A123545, A338511, A338583, A338584, A338593, A338604.

a(n) = A338604(n) - A338593(n).

A338604 Number of unlabeled connected graphs with n edges with degree >= 3 at each node.

Original entry on oeis.org

1, 0, 1, 3, 5, 9, 27, 67, 198, 646, 2216, 8178, 32095, 132093, 568368, 2541506, 11762657, 56183633, 276288402, 1396172601, 7238931364

Offset: 6

Hugo Pfoertner, Nov 21 2020

			a(10)=5:
There are 5 graphs with 10 edges and degree >=3 at all nodes (see table in A123545):
The complete graph on 5 nodes, given by the edge list
[[1,2],[1,3],[1,4],[1,5],[2,3],[2,4],[2,5],[3,4],[3,5],[4,5]],
and 4 graphs on 6 nodes:
  [[1,3],[1,5],[1,6],[2,4],[2,5],[2,6],[3,5],[3,6],[4,5],[4,6]],
  [[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,4],[3,5],[3,6],[4,6]],
  [[1,3],[1,4],[1,6],[2,4],[2,5],[2,6],[3,5],[3,6],[4,6],[5,6]],
  [[1,3],[1,4],[1,5],[1,6],[2,4],[2,5],[2,6],[3,5],[3,6],[4,6]].
The first one has degree 3 or 4 at all nodes, but becomes disconnected by removing nodes 5 and 6 and their incident edges. It is therefore not 3-connected.
    .--5--.
   /  / \  \
  1--3   4--2
   \  \ /  /
    .--6--.
.
The complete graph on 5 nodes and the last 3 graphs with 6 nodes are all 3-connected. Thus A338511(10)=4, and by inclusion of the graph shown above a(10)=5.

Cf. A007112, A123545, A338511, A338593, A338594.

A342558 a(n) is the maximum number of distinct currents > 0 in a network of n one-ohm resistors with a total resistance of 1 ohm.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 12, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 1

Hugo Pfoertner and Rainer Rosenthal, May 26 2021

nonn

The resistor networks considered here correspond to multigraphs in which each edge is replaced by one or more one-ohm resistors, and in which there are two distinguished nodes, called poles, between which there is a total resistance of 1 ohm.
It was known that the smallest resistor network with all currents being distinct consists of 21 resistors, found by Duijvestin in 1978. This assumes that the network is planar and thus the analogy to the perfectly tiled squares exists, see A014530. For history and references see link to Stuart Anderson's website "SPSS, Order 21".
In 1983, A. Augusteijn and A. J. W. Duijvestijn  described networks in which the number of resistors in a network with distinct resistances was reduced to 20 by allowing the tiled square to be wrapped onto a cylinder. (see links to their publication and to Stuart Anderson's website "Simple Perfect Square-Cylinders")
For values of n greater than 21 increasingly numerous square divisions with a(n) = n exist so that a(n) = n holds for all n > 21 (see A006983).
In the present sequence, networks based on non-planar graphs are allowed, which makes it possible to find networks with a(n) = n also for n = 18 and n = 19.
In the range from n = 13 to n = 17, larger numbers of distinct currents are found than are possible with the methods for generating Mrs. Perkins's quilts, which naturally correspond to planar graphs.

			Examples for n <= 21 are given in the Pfoertner links. Visualizations of tilings corresponding to optimal networks for n <= 12 are given in the Mathworld "Mrs. Perkins's Quilt" link.

Stuart Anderson, Simple Perfect Squared Squares (SPSSs), Order 21 to 37 and higher orders.
Stuart Anderson, SPSS, Order 21.
Stuart Anderson, Simple Perfect Square-Cylinders (SPSCs); Order 20.
Stuart Anderson, Mrs Perkins's Quilt.
A. Augusteijn and A. J. W. Duijvestijn, Simple perfect square-cylinders of low order, Journal of Combinatorial Theory, Series B, Volume 35, Issue 3, December 1983, Pages 333-337
A. J. W. Duijvestijn, Simple perfect squared square of lowest order, Journal of Combinatorial Theory, Series B, Volume 25, Issue 2, October 1978, Pages 240-243
Ed Pegg Jr., List of solutions for the Mrs. Perkins's Quilt Square packing problem.
Hugo Pfoertner, Examples of networks of n one-ohm-resistors with total resistance of 1 ohm, maximizing the number of distinct currents through the single resistors, May 2021, Jan-Apr 2023
Hugo Pfoertner, Visualization of the resistor networks with n >= 13 using the Mathematica graph function, Jan-Apr 2023
Rainer Rosenthal, Cascade graphs for examples n <= 21, January 2023
Rainer Rosenthal, Network and semi-quilt visualizing a(18), January 2023
Eric Weisstein's World of Mathematics, Mrs. Perkins's Quilt.

Cf. A002962, A005670, A006983, A014530, A160911, A180414, A181340, A217156, A337517, A338593, A342556, A360030.

a(n) = n for n >= 18.

A338583 Number of unlabeled 3-connected nonplanar graphs with n edges.

Original entry on oeis.org

1, 2, 3, 10, 29, 94, 343, 1291, 5206, 22061, 96908, 439837, 2053916, 9841412, 48319944, 242857491, 1248629027, 6563581656, 35258560001, 193463945790

Offset: 9

Hugo Pfoertner, Nov 21 2020

Hugo Pfoertner, Illustrations of terms a(9) - a(12).

Cf. A002840, A006290, A337517, A338197, A338511, A338573, A338584, A338593, A338594, A338604.

a(n) = A338511(n) - A002840(n).

a(n) <= A338593(n). The difference A338584(n) = A338593(n)-a(n) are the counts of nonplanar connected graphs with minimum degree 3 at each node that are not 3-connected.

A338584 Number of unlabeled nonplanar connected graphs with n edges with minimum degree 3 at each node that are not 3-connected.

Original entry on oeis.org

1, 6, 28, 128, 558, 2421, 10675, 47810, 217572, 1006211

Offset: 13

Hugo Pfoertner, Nov 21 2020

Hugo Pfoertner, Illustration of terms a(13) and a(14).

Cf. A006290, A007112, A123545, A338583, A338593, A338594, A338604.

a(n) = A338593(n) - A338583(n).

cp's OEIS Frontend

A052448 Number of simple unlabeled n-node graphs of edge-connectivity 3.

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A338594 Number of unlabeled connected planar graphs with n edges with degree >= 3 at each node.

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A338604 Number of unlabeled connected graphs with n edges with degree >= 3 at each node.

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A342558 a(n) is the maximum number of distinct currents > 0 in a network of n one-ohm resistors with a total resistance of 1 ohm.

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A338583 Number of unlabeled 3-connected nonplanar graphs with n edges.

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A338584 Number of unlabeled nonplanar connected graphs with n edges with minimum degree 3 at each node that are not 3-connected.

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